Answer: (a) \(P=(8,21)\); (b) \(\sqrt{505}\), approximately \(22.5\).

We start with the main method. First find the line equation from the given points, then solve simultaneously with the second line.

(a) First find the gradient of the line through \((4,23)\) and \((-8,29)\):

\(m=\frac{29-23}{-8-4}=\frac6{-12}=-\frac12.\)

Using the point \((4,23)\), the equation is

\(y-23=-\frac12(x-4).\)

So

\(y=-\frac12x+25.\)

At the point of intersection with \(y=2x+5\),

\(-\frac12x+25=2x+5.\)

Hence

\(20=\frac52x,\)

so

\(x=8.\)

Then

\(y=2(8)+5=21.\)

Therefore

\(P=(8,21).\)

(b) The distance of \(P\) from the origin is

\(\sqrt{8^2+21^2}=\sqrt{64+441}=\sqrt{505}.\)

This completes the solution and gives the required result.